The ratio S/√
B is proportional to the absolute number of events passing the cuts at a given integrated luminosity and is mainly determined by the signal cross section (Figure 5.12a), whereas the selection efficiency plays a minor role in the given parameter range. However, S/√
B overestimates the expected signal significance strongly and the binomial significance Z0 gives a more realistic estimate. In its calculation the system-atic uncertainty of 25 % (cf. Section 5.1.4) as well as the statistical uncertainty of the background estimate have been taken into account.
One can translate the expected signal significance into the required luminosity to reach the discovery significance of 5σ. The significance definition of Z0 cannot easily be inverted analytically and therefore a numerical method was used to calculate the contours of required luminosity given in Figure 5.13. Especially for high values of M1/2 the binomial significance requires unreasonable amounts of data (Figure 5.13b), because the number of selected signal events gets that low, that the significance is fully dominated by the systematic uncertainty of the background estimate. In this region of parameter space an increase of the collected integrated luminosity does not help anymore, but only better estimates of the Standard Model background allow for a discovery of the SUSY signal.
[GeV]
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Figure 5.14: Invariant mass distribution of the visible part of the hardest tau, τ±, with the two nearest (in ∆R) charged leptons, `, of the first or second generation. The black line (purple line) gives the distribution for the opposite-sign (same-sign) lepton pair, `+`− (`±`∓), plus the tau. The distribution is denoted by OS (SS). The green line shows the difference of the OS and SS distributions. The red histogram corresponds to the correct τ±`+`− combination, i.e. all three leptons stem from the same τe1 decay. A Gaussian (blue line) has been fitted to the OS−SS distribution.
still build the invariant mass of the`+`− pair with the visible part of the (hadronically decaying) τ. Here and in the following `± denotes e± or µ±. One then expects a kinematic endpoint in the invariant mass distribution, which should lie at the true stau LSP mass.
It has already been shown in Section 5.1.2 that (nearly) background-free samples of BC 1 events can be selected with efficiencies of roughly 20 %. Therefore, Standard Model backgrounds need not to be considered in the following investigations. However the combinatorial background due to wrong combinations of `+`−τhad selections turns out to be sizable. The black line in Figure 5.14 (labelled OS for opposite sign) shows the invariant mass distribution of the `+`− pair combined with the visible part of the tau.
The histogram contains 50 000 BC 1 signal events simulated at a centre-of-mass energy of √
s = 7 TeV. The red histograms includes only such combinations where the correct combination was chosen,i.e. all three particles belong to the same stau decay.
Note, that no detector simulation is employed here. Electrons and muons are only selected if their transverse momentum pT is larger than 7 GeV and |η| <2.5. Hadroni-cally decaying taus are identified if their visible momentum exceeds pvisT >10 GeV and
|ηvis| < 2.5. In general more than one τ±`+`− combination exists per event. In fact one would be fully dominated by wrong combinations, if all such combinations would be taken into account. This means it is crucial to develop a method having a high probabil-ity to select the correct combinations. In principle two correct combinations may exist in a single event, if both taus stemming from the twoτe1-LSPs decay hadronically. However, it is quite unlikely that all electrons, muons and taus in the event can be reconstructed and therefore only one combination is kept per event at maximum. As an additional cut only those selected combinations are used, where the distance in ∆R between both
leptons and the tau is smaller than 1.5. In principle this cut may distort the invariant mass spectrum, especially for very high stau masses. Both options with and without this cut have been checked and both give nearly identical results in the precision of the estimated stau mass. A detailed discussion of the combinatorial backgrounds and a comparison of different combination schemes is given in Appendix D.
5.3.1 Stau mass estimation
In order to reduce the combinatorial backgrounds at least statistically and thus to sharpen the kinematic endpoint of the invariant mass, one can combine the hardest tau with the nearest same sign (SS) lepton pair, `±`±. The respective invariant mass distribution is given by the purple line in Figure 5.14 (labelled SS). After subtraction of the same signτ `±`± distribution (purple line) from the opposite signτ `+`−distribution (black line) one obtains the OS−SS distribution given by the green line.
The OS−SS invariant mass distribution follows much closer the distribution that arises from the correct τ `+`− triplet (red histogram in Figure 5.14) than the original OS distribution. Without the cut on the angular distance between the two leptons and the tau the same sign distribution shows a long tail at high invariant masses, which leads to an over-subtraction at high masses. This can be explained by the fact, that one of the same sign leptons mostly stems from the other stau decay in the event or another source and therefore often has a larger angle to the first lepton leading to higher invariant masses.
The resulting OS−SS distribution has an endpoint near the true endpoint at the simulated eτ1 mass of 148 GeV. However, it is difficult to define in measured data, where the endpoint exactly lies as always a few entries will remain with higher invariant masses.
One therefore needs to find an observable which is sensitive to the stau mass. Here a Gaussian distribution is fitted to the OS−SS distribution and the value calculated, where it drops to 10% of its maximum (marked by a star in Figure 5.14). The fit range is crucial and has been determined in an iterated Gaussian fit, which starts with the maximum bin position and the root mean square (RMS) of the histogram and uses the range µ−p
2 ln(2)/2·σ to 200 GeV, whereµis the mean of the previous Gaussian fit and σ2 its variance. The effect of the fit range needs to be treated as a systematic uncertainty of the method. Although this observable lies below the simulated stau mass of 148 GeV, one can use it to estimate the true mass as long as there is a clear and known correlation between the two. Such a procedure has already been successfully demonstrated on simulated Monte Carlo samples in the context of sparticle mass measurements e.g. in reference [67, 102].
Other functions for the fit have been tested as well, like the log-normal distribution or a Gaussian distribution with an additional offset. From all the tested functions the simple Gaussian with the above mentioned fit range turned out the give the most stable fit results, while describing the invariant mass distribution sufficiently well. Similarly, other definitions of the observable were used, like the mean of the Gaussian or its upper
stau mass [GeV]
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Figure 5.15: Stau mass sensitive observable versus true stau mass (see text for definition) for the scenarios presented in Section 5.2. Different colours of the points correspond to different intervals of theχe01(NLSP) mass. (a) includes all parameter points, where 10 000 signal events were simulated each. (b) shows estimates for an integrated luminosity of 5 fb−1 at √
s = 7 TeV including event selection cuts. Only scenarios where at least 150 events pass the cuts are used in (b). The error bars show to what precision the estimated stau mass can be measured. The errors correspond to statistical fluctuations and are estimated as described in the text.
half maximum value. Again, the definition finally used showed the best correlation with the true stau mass.
Indeed a clear correlation between the aforementioned observable and the true eτ mass is observed in Figure 5.15a. Here the stau LSP scenarios of the parameter scans from Section 5.2 were taken. 10 000 signal events were simulated for each scenario and the observable determined from the OS−SS invariant mass distribution as described above. The different marker styles in Figure 5.15a correspond to different masses of the χe01, which is the next-to-lightest supersymmetric particle (NLSP) here. There is only a small systematic dependency of the estimated mass on theχe01 mass. For example, for a stau mass of 120 GeV the observable can increase from roughly 100 GeV to 140 GeV, if the χe01 mass increases from 120 GeV to 240 GeV. This is expected because a heavier χe01 leads to a harder tau from theχe01 decay.
One can use Figure 5.15a to translate the observable to the true stau mass. Such an analysis is even possible in a limited way with early LHC data, as can be seen in Fig-ure 5.15b, where the observable is again plotted against the true stau mass. Here event selection cuts were applied and only scenarios were included, where at least 150 events in 5 fb−1 pass the event selection cuts. Otherwise, there would not be enough statistics for the mass reconstruction. The error bars correspond to the precision with which the observable can be measured assuming an integrated luminosity of 5 fb−1. Systematic uncertainties are not included here.
Figure 5.16: OS−SS distribution as in Fig-ure 5.14, but here including event selec-tion cuts and randomly selecting events corresponding to an integrated luminos-ity of 5 fb−1, i.e. uncertainties and fluc-tuations correspond to those expected for 5 fb−1at√
s= 7 TeV. The star marks the
value of the chosen observable. ml l τ [GeV]
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The statistical uncertainties in Figure 5.15b were estimated in the following way.
Out of 10 000 simulated signal events, events have been randomly chosen to get a sub-sample corresponding to an integrated luminosity of 5 fb−1. This procedure was repeated 100 times for each point to obtain different samples. The observables of these sub-samples follow a Gaussian distribution, where its width corresponds to the statistical uncertainties in Figure 5.15b. Figure 5.14 shows an example of one sub-sample for the BC 1 scenario.
From the above investigations one can conclude that rough estimates of the stau LSP mass in BC 1-like scenarios should be possible with an integrated luminosity of 5 fb−1 at√
s = 7 TeV. An interpretation of candidate events within the assumed model will be feasible with a mass resolution of about 20 GeV depending on the actual eτ mass. More detailed studies of the systematic effects on the mass determination have to be done once a BSM signal has been established and their specific event topologies are known.